Rounding PiDate: 06/01/99 at 11:37:02 From: Sam Gibson Subject: Can Pi be rounded to 3.0? I have been challenged to prove that the value of Pi cannot be rounded down to 3.0. I balked at this, thinking that it is obvious that it would make all calculations needing Pi incorrect, but I can find nothing that would say that this is improper. At the same time, I find nothing that says that Pi cannot be rounded to 5, either. Can you help? Thanks! Sam Gibson Date: 06/01/99 at 12:09:26 From: Doctor Peterson Subject: Re: Can Pi be rounded to 3.0? Hi, Sam. I'm not sure of the context of this question. Any number can be rounded to any precision you want; you can round pi to the nearest ten if you want, and you'll get zero. Rounding it to the nearest unit, giving 3, makes perfectly good sense if that is what you want to do. The real question is, what is lost if you round pi down to 3? You're reducing its value by .14/3.14 = 4.5%, so any calculations you make will have that much error; but for many purposes that would be perfectly acceptable. Whenever we work with pi we are rounding it to some number of digits, so all such calculations are incorrect. The only issue is how much accuracy we need for a particular application. I hope that helps. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/01/99 at 12:21:44 From: Doctor Rob Subject: Re: Can Pi be rounded to 3.0 You can round pi to anything you want, but you won't have pi when you are done. If you try to use that number to find the circumference of a circle with a given diameter, you will get answers that are not accurate. Actually, I think what the challenger wants is for you to prove that pi > 3.1, so that when rounded to two significant figures, you don't get 3.0. This can be done as follows. Start with a regular hexagon inscribed in a circle with diameter d. The perimeter of the hexagon is p = 3*d, so Pi > p/d = 3. Now bisect each side of the hexagon, and connect the points where the perpendicular bisectors meet the circle to the vertices of the hexagon, thus forming a dodecagon (regular 12-sided polygon) also inscribed in the same circle. You can show that the perimeter of this polygon is p = 3*(sqrt(6)-sqrt(2))*d, so pi > p/d = 3.1058... > 3.1, which is what you wanted. The small isosceles triangles formed will have base d/2, base angles of 15 degrees (or Pi/12 radians), and vertex angle of 150 degrees (or 5*Pi/6 radians). You can use trigonometry to find the lengths of the other two sides of the triangle (the Law of Sines comes to mind), once you know the sine of 15 degrees, which is [sqrt(6)-sqrt(2)]/4, and that the sine of 150 degrees is 1/2. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 06/01/99 at 12:40:32 From: Sam Gibson Subject: Re: Can Pi be rounded to 3.0 Hi Dr. Peterson, The purpose of this little experiment was to show that there is an error in the Bible. In 1 Kings 7:23, a circumference of a circular object is given as 30 while the diameter is given as 10. Obviously, when using the equation 2Pr (where P = pi) to find the circumference this would make pi equal to 3. I was wondering if it is considered proper to round pi down to 3. I understand that pi is always rounded no matter what, but is it proper to round it that much? Thanks. Sam Gibson Date: 06/01/99 at 12:51:06 From: Sam Gibson Subject: Re: Can Pi be rounded to 3.0 Thank you very much. I am actually doing this to show that there is an error in the Bible at 1 Kings 7:23 and have run into a person who claims that there is no error here and that it is proper to round the value of pi to be 3.0. The verse claims that there is a circular object that has a circumference of 30 and a diameter of 10. You have shown that this can not be. Thanks again. Sincerely, Sam Gibson Date: 06/01/99 at 16:08:16 From: Doctor Rick Subject: Re: Can Pi be rounded to 3.0 Hi, Sam. I beg to differ. The question that you asked, and that Dr. Rob answered, is not the same question that you have now stated. Dr. Rob has not proved your claim at all. The Bible does not state that pi = 3.0. It states that a particular object (the circular basin in front of the Jerusalem Temple) had a diameter of 10 cubits and a circumference of 30 cubits. So the correct question is not, "Is it proper to round pi to 3.0?" but "Is it proper to round the circumference of this circle to 30 cubits?" Or better, "Are a diameter of 10 cubits and a circumference of 30 cubits consistent within reasonable measurement error?" We do not know the precision of the measuring instruments used to measure the diameter and circumference of this circle. But here is what I would naturally understand if I saw this figure in a scientific journal: in the absence of an explicit indication of precision, the absence of a tenths digit implies that the figure is accurate to the nearest 1 cubit - that is, plus or minus 0.5 cubit. So let's suppose that the diameter was measured, or specified in the design, to be 10 cubits plus or minus 0.5 cubit. Then the actual circumference would be in the range from 9.5 pi to 10.5 pi, or 29.8 to 32.98 cubits. If we make the same assumption about the precision of the circumference measurement, we get a range of 29.5 to 30.5 cubits. Notice that the two ranges have considerable overlap. There is therefore no inconsistency between the diameter and the circumference as reported in the Bible. If you have further questions of this nature, I hope you will be careful in stating them so that the facts are properly represented. Doctor Math does not want to have inaccurate statements attributed to him/her! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 06/01/99 at 16:44:47 From: Sam Gibson Subject: Re: Can Pi be rounded to 3.0 Doctor Rick, I am not quite sure I am understanding you. I do not think that I misrepresented the question at all. What I asked is simply, can pi ever be expressed as being equal to 3.0. The Bible makes this claim and it is mathematically incorrect. You wrote: "in the absence of an explicit indication of precision, the absence of a tenths digit implies that the figure is accurate to the nearest 1 cubit - that is, plus or minus 0.5 cubit." Why? The Bible does not say approximately or about 10 cubits. It states that it is 10 cubits. I am not sure why we could take that to mean anything other than 10 cubits which is what we would need to do if we accept a range from 9.5 cubits to 10.5 cubits. "So let's suppose that the diameter was measured, or specified in the design, to be 10 cubits plus or minus 0.5 cubit. Then the actual circumference would be in the range from 9.5 pi to 10.5 pi, or 29.8 to 32.98 cubits." First, we would need to understand why this is acceptable to do when we have been given a figure of 10. "If we make the same assumption about the precision of the circumference measurement, we get a range of 29.5 to 30.5 cubits. Notice that the two ranges have considerable overlap. There is therefore no inconsistency between the diameter and the circumference as reported in the Bible." Same problem as above. The figure given is 30, not "approximately 30" or "about 30." "If you have further questions of this nature, I hope you will be careful in stating them so that the facts are properly represented. Doctor Math does not want to have inaccurate statements attributed to him/her!" I probably won't have future questions like this. Again, the question that I asked was: "Is it proper to round pi to 3.0?" This would need to be the case if we understand the diameter to be 10 and the circumference to be 30. Thanks again. Sam Gibson Date: 06/01/99 at 19:22:51 From: Doctor Rick Subject: Re: Can Pi be rounded to 3.0 Hi again, Sam. No, you didn't misrepresent the question, but you didn't give enough information for either Dr. Peterson or Dr. Rob to interpret the phrase "can be rounded." Both said that pi can be rounded however you like. Dr. Rob made a reasonable guess at what criterion you might have in mind, supposing that you needed to prove that pi > 3.1. Re-read Dr. Peterson's answer, please: "I'm not sure of the context of this question. Any number can be rounded to any precision you want; you can round pi to the nearest ten if you want, and you'll get zero. Rounding it to the nearest unit, giving 3, makes perfectly good sense if that is what you want to do. The real question is, what is lost if you round pi down to 3? You're reducing its value by .14/3.14 = 4.5%, so any calculations you make will have that much error; but for many purposes that would be perfectly acceptable. Whenever we work with pi we are rounding it to some number of digits, so all such calculations are incorrect! The only issue is how much accuracy we need for a particular application." We need to start from the actual question posed to you in order to decide what are the appropriate criteria - how accurate we need to be in this particular application. Then we can see whether approximating pi as 3.0 meets these criteria. >What I asked is simply, can pi ever be expressed as being equal to >3.0. The Bible makes this claim and it is mathematically incorrect. The Bible does not make this claim. If we are going to be mathematically precise, the Bible never says that there is such a thing as pi. The figure of 3.0 is not found there. It simply reports two measurements which imply a value of pi. Therefore in my analysis, I started from the measurements as given and derived reasonable precision criteria from them. This information was not available in the original question. >You wrote: >in the absence of an explicit indication of precision, the absence of >a tenths digit implies that the figure is accurate to the nearest 1 >cubit - that is, plus or minus 0.5 cubit. > >Why? The Bible does not say approximately or about 10 cubits. It >states that it is 10 cubits. I am not sure why we could take that to >mean anything other than 10 cubits which is what we would need to do >if we accept a range from 9.5 cubits to 10.5 cubits. Look back at Dr. Peterson's answer. Every measurement we ever make is an approximation. I don't know your background, but I studied physics and before we had our first laboratory, we learned about how to write our measurements so that they would accurately reflect the precision of our measuring instruments. If the smallest markings on my ruler are millimeters, then I would not write a measurement as 11.423 millimeters; the closest I could measure with that ruler would be plus or minus 0.5 millimeter, so I might write 11.5 mm +- 0.5 mm or even just 11 +- 0.5 mm. Such a measurement precision is part of every measurement, whether it is written or not. >> So let's suppose that the diameter was measured, or specified in >> the design, to be 10 cubits plus or minus 0.5 cubit. Then the >> actual circumference would be in the range from 9.5 pi to 10.5 pi, >> or 29.8 to 32.98 cubits. > >First, we would need to understand why this is acceptable to do when >we have been given a figure of 10. See my last paragraph. I don't know what the actual precision of the measurements was; I can only make a reasonable guess. I know it was not exactly 10, because as I said, no real-world measurement is exact. >> If we make the same assumption about the precision of the >> circumference measurement, we get a range of 29.5 to 30.5 cubits. >> Notice that the two ranges have considerable overlap. There is >> therefore no inconsistency between the diameter and the >> circumference as reported in the Bible. > >Same problem as above. The figure given is 30, not "approximately 30" >or "about 30". > >> If you have further questions of this nature, I hope you will be >> careful in stating them so that the facts are properly represented. >> Doctor Math does not want to have inaccurate statements attributed >> to him/her! > >I probably won't have future questions like this. Again, the question >that I asked was: "Is it proper to round pi to 3.0?" This would need >to be the case if we understand the diameter to be 10 and the >circumference to be 30. I hope I have made myself clearer this time. Under some conditions, as Dr. Peterson said, 5% accuracy is enough. My analysis showed that, with some guesses about the precision of the measurements, it seems that 5% may indeed have been enough in the context. In any case, it's not right to take Dr. Rob's answer as "proof" of your claim. Dr. Rob was answering a different assumed question, namely, how can you prove that pi is greater than 3.1. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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